solve-x-x-3-5-

Question Number 109193 by mr W last updated on 21/Aug/20

s o l v e

x^{x^{3}} = 5

Commented by aurpeyz last updated on 21/Aug/20

x^{x^{3}} = 2^{2}

x^{3} = 2

x {\underset{}{=}}^{3} \sqrt{2}

Commented by aurpeyz last updated on 21/Aug/20

i j u s t t r i e d . p l s c o r r e c t m e

Commented by mr W last updated on 21/Aug/20

i f x^{3} = 2 t h e n x = 2, i m p o s s b l e .

Commented by mr W last updated on 21/Aug/20

i h a v e c h a n g e d t h e q u e s t i o n t o

x^{x^{3}} = 5

Commented by aurpeyz last updated on 21/Aug/20

i w r o t e c u b e r o o t o f 2

Commented by aurpeyz last updated on 21/Aug/20

o k a y s i r . i w i l l t r y

Commented by mr W last updated on 21/Aug/20

y o u w r o t e

x^{x^{3}} = 2^{2}

i f x^{3} = 2, t h e n y o u m e a n a l s o x = 2 .

Commented by aurpeyz last updated on 21/Aug/20

y e s . i a m w r o n g . i f t h e i n d e c e s a r e e q u a l . t h e b a s e

w i l l b e e q u a l .

Answered by Dwaipayan Shikari last updated on 21/Aug/20

x^{3} l o g x = l o g 5

3 l o g x e^{3 l o g x} = 3 l o g 5

3 l o g x = W_{0} (3 l o g 5)

x = e^{\frac{W_{0} (3 l o g 5)}{3}} = 1.5459 . .

Commented by mr W last updated on 21/Aug/20

g o o d!

Answered by Aziztisffola last updated on 21/Aug/20

\ln (x^{x^{3}}) = \ln (5) \Leftrightarrow x^{3} \ln (x) = \ln (5)

\Leftrightarrow \ln (x) e^{3 \ln (x)} = \ln (5)

\Leftrightarrow 3 \ln (x) e^{3 \ln (x)} = 3 \ln (5)

W (3 \ln (x) e^{3 \ln (x)}) = W (3 \ln (5))

3 \ln (x) = W (3 \ln (5))

x = e^{\frac{W (3 \ln (5))}{3}}

Commented by aurpeyz last updated on 24/Aug/20

Pls what is the w or w_{0} being inserted ?

Commented by Aziztisffola last updated on 24/Aug/20

The Lambert W function; the invers

function of f : x ∣ \to {x e}^{x}

W = f^{- 1}

Commented by aurpeyz last updated on 25/Aug/20

w o w . t h a t w a s w h y y o u m u l t i p l i e d t h r o u g h

b y 3 ?

Answered by 1549442205PVT last updated on 22/Aug/20

Put x^{3} = y \Rightarrow x =^{3} \sqrt{y} \Rightarrow^{3} {(\sqrt{y})}^{y} = 5

yln (^{3} \sqrt{y}) = \ln 5 \Leftrightarrow \frac{y}{3} lny = \ln 5

ylny = 3 \ln 5 \Leftrightarrow y = e^{\frac{3 \ln 5}{y}} \Leftrightarrow \frac{1}{y} . e^{\frac{3 \ln 5}{y}} = 1

Put \frac{1}{y} = z we get {ze}^{z . 3 \ln 5} = 1

\Leftrightarrow z . {3 \ln 5 \ln 5 e}^{z . 3 \ln 5} = 3 \ln 5

\Rightarrow z . 3 \ln 5 = W_{0} (3 \ln 5) = 1.306568 . .

\Rightarrow z = 1.306568 / (3 \ln 5) = 0.27060 .

\Rightarrow y = 1 / z = 3.695417

\Rightarrow x =^{3} \sqrt{y} = 1.5460